In their
article "On analytic iteration"
Erdös & Jabotinsky [1] stated
1960, that "the
function ez-1 was shown by I.N.Baker to have no real non-integer
iterates" and cited I.N.Bakers article of 1958.

However, I.N.Baker [2] deals with this question
in terms of "radius of convergence", and the focus of his proof is,
that "the
radius of convergence isnonzero iff the iterator h is integer"
(rough translation).

The latter can
be shown by numerical examples. However, although the simple inspection seconds
this strongly, the zero-radius of convergence does not mean, that to such a powerseries
a value could not be assigned principally. If we study powerseries in the well
established context of summation of divergent series (see for instance [3] or
[4]), we are no more lost in space: we may apply Cesaro-/Euler- or
Borel-summation to assign values to a divergent series, if their terms oscillate
in sign and "diverge not too strong" (L. Euler).

Thus, if we can
show, that the growthrate of the terms of the occuring series "is not too
strong", then we may assign values to continuous U-tetration as well, based on those concepts of divergent
summation.

From empirical
evidence it seems, that the growthrate of terms is roughly hypergeometric (the
logarithms of their absolute values increase roughly with their index) and may
be Euler- or Borel-summable.

Adding the
concept of divergent summation seems to fill the gap, which is the reason for
the common expression, that continuous iteration / tetration were not existent/meaningful.

The precise
growthrate of terms is not yet determined. However, it seems, that the forward
differences (along columns) for all documented columns tend to be roughly
equal for the fractional heights, perhaps depending on some index-shift.

Table 3: forward differences 1'st
order of log10
of absolute values of terms

h=1

h=1+1e-10

h=1+1e-9

h=1+1e-8

h=1+1e-7

h=1+1e-6

h=1+1e-5

h=1+1e-4

h=1.001

h=1.01

h=1.1

-0.3010

-0.3010

-0.3010

-0.3010

-0.3010

-0.3010

-0.3010

-0.3010

-0.3006

-0.2967

-0.2596

-0.4771

-0.4771

-0.4771

-0.4771

-0.4771

-0.4771

-0.4771

-0.4771

-0.4765

-0.4707

-0.4164

-0.6021

-0.6021

-0.6021

-0.6021

-0.6021

-0.6021

-0.6021

-0.6020

-0.6012

-0.5935

-0.5229

-0.6990

-0.6990

-0.6990

-0.6990

-0.6990

-0.6990

-0.6990

-0.6989

-0.6980

-0.6889

-0.6065

-0.7782

-0.7782

-0.7782

-0.7782

-0.7782

-0.7782

-0.7781

-0.7780

-0.7770

-0.7670

-0.6755

-0.8451

-0.8451

-0.8451

-0.8451

-0.8451

-0.8451

-0.8451

-0.8450

-0.8437

-0.8313

-0.7271

-0.9031

-0.9031

-0.9031

-0.9031

-0.9031

-0.9031

-0.9031

-0.9030

-0.9019

-0.8906

-0.7788

-0.9542

-0.9542

-0.9542

-0.9542

-0.9542

-0.9542

-0.9543

-0.9543

-0.9550

-0.9590

-0.8690

-1.0000

-1.0000

-1.0000

-1.0000

-1.0000

-1.0000

-0.9998

-0.9982

-0.9825

-0.8668

-0.7196

-1.0414

-1.0414

-1.0414

-1.0414

-1.0414

-1.0414

-1.0412

-1.0394

-1.0222

-0.9008

-0.5827

-1.0792

-1.0792

-1.0792

-1.0792

-1.0793

-1.0805

-1.0925

-1.2336

-0.8386

0.0324

-0.1389

-1.1139

-1.1139

-1.1139

-1.1139

-1.1133

-1.1074

-1.0509

-0.6170

-0.2797

-0.5184

-2.0779

-1.1461

-1.1461

-1.1460

-1.1447

-1.1323

-1.0254

-0.5563

0.0468

0.2656

0.3480

2.0140

-1.1761

-1.1763

-1.1784

-1.1993

-1.4808

-0.7138

-0.1371

-0.0290

-0.0191

-0.0421

-0.3110

-1.2041

-1.2060

-1.2233

-1.4655

-0.3353

-0.1213

-0.2037

-0.2106

-0.2069

-0.1643

0.2332

-1.2304

-1.1489

-0.7307

0.3767

0.4962

0.4017

0.3932

0.3920

0.3885

0.3550

0.0825

-1.2553

-1.2044

-1.0804

-1.0230

-1.0150

-1.0142

-1.0139

-1.0119

-0.9923

-0.8322

-0.1630

-1.2788

0.6344

1.1150

1.2122

1.2232

1.2243

1.2242

1.2223

1.2033

1.0491

0.4450

-1.3010

-0.4566

-0.4606

-0.4610

-0.4610

-0.4611

-0.4611

-0.4617

-0.4676

-0.5315

-0.7423

-1.3222

0.7240

0.7245

0.7246

0.7246

0.7246

0.7246

0.7253

0.7317

0.8007

1.0610

-1.3424

-0.0674

-0.0674

-0.0674

-0.0674

-0.0674

-0.0675

-0.0678

-0.0708

-0.1022

-0.5937

-1.3617

0.3854

0.3854

0.3854

0.3854

0.3854

0.3854

0.3858

0.3894

0.4262

0.9646

-1.3802

0.1877

0.1877

0.1877

0.1877

0.1877

0.1877

0.1874

0.1848

0.1590

-0.1358

-1.3979

0.1763

0.1763

0.1763

0.1763

0.1763

0.1764

0.1767

0.1800

0.2127

0.5605

-1.4150

0.4115

0.4115

0.4115

0.4115

0.4115

0.4115

0.4112

0.4085

0.3815

0.1213

-1.4314

-0.0209

-0.0209

-0.0209

-0.0209

-0.0209

-0.0209

-0.0205

-0.0166

0.0214

0.3528

-1.4472

0.6469

0.6469

0.6469

0.6469

0.6469

0.6468

0.6465

0.6429

0.6088

0.3274

-1.4624

-0.2788

-0.2788

-0.2788

-0.2788

-0.2788

-0.2787

-0.2781

-0.2719

-0.2138

0.1873

-1.4771

0.9499

0.9499

0.9499

0.9499

0.9499

0.9499

0.9493

0.9434

0.8882

0.5220

-1.4914

-0.8630

-0.8630

-0.8630

-0.8630

-0.8629

-0.8627

-0.8604

-0.8379

-0.6595

0.0169

-1.5051

1.5811

1.5811

1.5811

1.5811

1.5810

1.5808

1.5785

1.5563

1.3804

0.7314

-1.5185

-0.5965

-0.5965

-0.5965

-0.5965

-0.5965

-0.5966

-0.5979

-0.6113

-0.7732

-0.2050

-1.5315

1.3610

1.3610

1.3610

1.3610

1.3610

1.3612

1.3625

1.3761

1.5403

0.9955

-1.5441

-0.1747

-0.1747

-0.1747

-0.1747

-0.1747

-0.1747

-0.1753

-0.1807

-0.2391

-0.6245

-1.5563

0.9842

0.9842

0.9842

0.9842

0.9842

0.9842

0.9848

0.9905

1.0511

1.4578

-1.5682

0.0530

0.0530

0.0530

0.0530

0.0530

0.0529

0.0526

0.0489

0.0107

-0.8305

-1.5798

0.7994

0.7994

0.7994

0.7994

0.7994

0.7995

0.7999

0.8038

0.8442

1.7059

-1.5911

0.2193

0.2193

0.2193

0.2193

0.2193

0.2192

0.2189

0.2160

0.1855

-0.2594

-1.6021

0.6738

0.6738

0.6738

0.6738

0.6738

0.6738

0.6741

0.6774

0.7101

1.1756

-1.6128

0.3571

0.3571

0.3571

0.3571

0.3571

0.3570

0.3568

0.3541

0.3272

-0.0064

-1.6232

0.5741

0.5741

0.5741

0.5741

0.5741

0.5741

0.5744

0.5773

0.6068

0.9620

-1.6335

0.4801

0.4801

0.4801

0.4801

0.4801

0.4801

0.4798

0.4773

0.4517

0.1666

-1.6435

0.4863

0.4863

0.4863

0.4863

0.4863

0.4863

0.4866

0.4895

0.5180

0.8266

-1.6532

0.5959

0.5959

0.5959

0.5959

0.5959

0.5959

0.5957

0.5931

0.5674

0.3039

-1.6628

0.4025

0.4025

0.4025

0.4025

0.4025

0.4025

0.4028

0.4058

0.4350

0.7251

-1.6721

0.7098

0.7098

0.7098

0.7098

0.7098

0.7098

0.7095

0.7068

0.6798

0.4220

-1.6812

0.3166

0.3166

0.3166

0.3166

0.3166

0.3167

0.3170

0.3202

0.3518

0.6408

-1.6902

0.8264

0.8264

0.8264

0.8264

0.8264

0.8264

0.8261

0.8231

0.7933

0.5291

-1.6990

0.2227

0.2227

0.2227

0.2227

0.2227

0.2228

0.2232

0.2269

0.2628

0.5654

-1.7076

0.9512

0.9512

0.9512

0.9512

0.9512

0.9511

0.9508

0.9472

0.9129

0.6303

-1.7160

0.1126

0.1126

0.1126

0.1126

0.1126

0.1127

0.1131

0.1177

0.1612

0.4937

-1.7243

1.0918

1.0918

1.0918

1.0918

1.0918

1.0918

1.0913

1.0869

1.0448

0.7291

-1.7324

-0.0280

-0.0280

-0.0280

-0.0280

-0.0280

-0.0279

-0.0273

-0.0212

0.0362

0.4219

-1.7404

1.2624

1.2624

1.2624

1.2624

1.2624

1.2623

1.2617

1.2557

1.1996

0.8285

-1.7482

-0.2326

-0.2326

-0.2326

-0.2326

-0.2326

-0.2325

-0.2315

-0.2218

-0.1346

0.3463

-1.7559

1.4963

1.4963

1.4963

1.4963

1.4963

1.4962

1.4952

1.4856

1.3997

0.9317

-1.7634

-0.6338

-0.6338

-0.6338

-0.6338

-0.6337

-0.6335

-0.6310

-0.6068

-0.4184

0.2628

-1.7709

1.9261

1.9261

1.9261

1.9261

1.9261

1.9258

1.9233

1.8993

1.7119

1.0425

-1.7782

-0.8967

-0.8967

-0.8967

-0.8967

-0.8967

-0.8972

-0.9019

-0.9523

-1.5908

0.1656

-1.7853

2.2169

2.2169

2.2169

2.2169

2.2169

2.2174

2.2221

2.2726

2.9122

1.1667

-1.7924

-0.3036

-0.3036

-0.3036

-0.3036

-0.3036

-0.3037

-0.3050

-0.3176

-0.4694

0.0446

-1.7993

1.6509

1.6509

1.6509

1.6509

1.6509

1.6510

1.6523

1.6650

1.8179

1.3141

It is
interesting to see the ratio of consecutive coefficients graphically.

Formal powerseries,
whose consecutive coefficients approach a constant ratio have a finite radius
of convergence. Formal powerseries, whose radius of convergence is infinite
must have a hypergeometric rate of decrease, as for instance the formal powerseries
for exp(x). Then there are formal powerseries whose ratio
increases with the index, for instance the Eulerian powerseries 1!x  2!x^2+3!x^3- Such formal powerseries have convergence-radius
zero.

As I.N. Baker
has shown, all fractional iterates off(x)=exp(x)-1 have convergence-radius zero. But they show an interesting pattern of
the increase of the ratio of concecutive oefficients. In the following plot I
show the log10 of the ratios (actually of the absoulute
value) in a double-logarithmic scale.

The blue line
represents the f°1(x) at the integer height 1; the magenta line that of f°1+µ(x), with µ=10^-10 which means a very near-integer-iterate.

The blue line
decreases linearly with the index k, which reflects, that the ratio r(k)of two coefficients of the powerseries for exp(x)-1 at the index k is just (k-1)!/k! = 1/k and thus the represented function
is entire. The same ratios in the fractional iterate follow that of the integer
iterate, but then begin to deviate and show oscillating behaviour. Additionally,
that oscillation is also overlaid by an increase which seems to be linear in
this scale.

If we look at a
fractional iterate farther away from the integer-iterate we get the next plot,
where now the green line indicates f°1+µ(x) where µ=10^-1. We get the same pattern, only that
the point of deviation and the beginning of oscillation is at an earlier index.

This suggests
another, surprising, interpretation of the exponential-series itself: the powerseries
of integer-iterates are only limits, where the index of beginning of increase
is shifted to infinity  and thus does no more appear

For a
computation of f°0.5(x) see the example at [6], where I computed the series using
Euler-summation of order 2.5.

Additional
remark:

If f(x) = exp(x) 1
is generalized to arbitrary bases fb(x) = bx-1, the terms
of the powerseries for exp(x)-1 must be scaled by powers of u=log(b).
If u<>1
(or is left symbolically) an eigensystem-decomposition of the associated
triangular matrix Ub can be
performed and the diagonal (and eigenvalues) are the powers of u. An
arbitrary height h
of iteration can then be determined by substituting u by the h'th power of u in
the set of eigenvalues. Then the fractional power of Ubh
can be symbolically described and in its second column we have the coefficients
for the associated powerseries for the continuous iterated fb°h(x).

Here are the
first four terms of the powerseries of height h computed via the symbolic eigensystem-decomposition
of the U-tetration-matrix S2b (see other articles in
tetration-index [5])

We see, that,
if u=1
, which is the case when fb(x) = exp(x)-1, the coefficients
at x
have zeros in the denominator, and produce a singularity, if not uh
is a positive integer power of u and numerators cancel the denominators. By
shift of the parameter x this is convertible to the usual-tetration
with base eta=e^(1/e)
and answers then the same question for this version of tetration.

Thus the
eigensystem-decomposition may be a good approach to go to study the general
question of integer vs fractional iteration in more detail.